What's the difference between those words with regard to a public showing, as of goods or works of art? Can these be used interchangeably?
Both "exhibit" and "exposition" are marked as Americanisms by Adrienne's English-French Dictionary, Ed. 1991.
EXPOSITION
A public show or exhibition (M-W)
Bertha, by this time a writer and critic, had attended an exposition in Cologne where she had seen Gropius's work enthusiastically received.
Gustav Mahler: A Life in Crisis
Werkbund Exposition, 1914. Deutz machine factory and office building. Court elevation, showing glass-walled offices & the covered terrace on the roof.; Walter Gropius; 1914
EXHIBITION
A public showing (as of works of art, objects of manufacture, or athletic skill) a one-man exhibition; an exhibition game (M-W)
An auto show, also known as a motor show or car show, is a public exhibition of current automobile models, debuts, concept cars, or out-of-production classics. It is attended by automotive industry representatives, dealers, auto journalists and car enthusiasts.
EXHIBIT
North American An exhibition: people flocked to the exhibit in record-breaking numbers
There he attended an exhibit of Nazi paraphernalia held on the campus of Tubingen University
The Pot Plan: Louie B. Stumblin and the War on Drugs
His one great success was a highly innovative glass house exhibited at the Cologne Werkbund exhibit in 1914.
Best Answer
In American English, originally the word "exhibit" as a noun was not applied to "goods or works of art" generally.
Instead, quoting Noah Webster's 1839 An American dictionary of the English language:
The 1904 Funk and Wangnalls A Standard Dictionary of the English Language does have a noun-meaning of "exhibit" of, among other meanings:
The current Webster's definition of EXHIBIT as a noun is:
So in the mid 1800's exhibition and exposition were interchangeable, and now all 3 are interchangeable.